The deal was first reported by Jon Morosi of Fox Sports.
Gyorko is coming off a season in which he hit .247 with 16 home runs and 57 RBIs for San Diego. He remains under team control for the next four years after inking a $35 million extension in April 2014 that includes a $13 million club option for 2020. He signed the extension after clubbing 23 home runs in 125 games for his rookie season.
The Cardinals will take on $26 million of Gyoroko's salary while the Padres will pay $6.2 million, according to USA Today's Bob Nightengale.
The left-handed hitting Jay, 30, has $6.85 million remaining on his contract before he becomes a free agent after the 2016 season. Jay hit .210 with one home run and 10 RBIs for the Cardinals last season, playing in only 79 games as he dealt with a wrist injury.
- Christopher Chavez